The present invention relates to internal combustion engines with multiple cylinders, of the type comprising:                at least one intake valve and at least one exhaust valve for each cylinder, each provided with respective return spring means which bias the valve towards a closed position, to control respective intake and exhaust conduits,        at least one camshaft, to actuate the intake and exhaust valves of the engine cylinders by means of respective tappets,        in which each intake valve has variable actuation, being actuated by the respective tappet, against the action of the aforesaid return spring means, by the interposition of hydraulic means including a pressurised fluid chamber, into which projects a pumping piston connected to the tappet of the intake valve,        said pressurised fluid chamber being able to be connected by means of a solenoid valve with an exhaust channel, in order to uncouple the variable actuation valve from the respective tappet and cause the rapid closure of the valve by effect of the respective return spring means,        electronic control means for controlling each solenoid valve in such a way as to vary the time and travel of opening of the variable actuation valves as a function of one or more operative parameters of the engine,        in which the aforesaid hydraulic means further comprise an actuator unit for each variable actuation valve, including an actuating piston slidably mounted in guide bushing,        said actuating piston facing a variable volume chamber communicating with the pressurised fluid chamber both through first communication means controlled by a check valve which allows only the passage of the fluid from the pressurised fluid chamber to the variable volume chamber, and through second communication means which allow the passage between the two chambers in both directions,        in which said hydraulic means further comprise hydraulic braking means able to cause a narrowing of said second communication means in the final phase of closure of the engine valve,        in which between the actuating piston of each variable actuation valve and the stem of the valve is interposed an auxiliary hydraulic tappet,        in which said auxiliary hydraulic tappet comprises:        a first bushing having an end wall in contact with one end of the stem of the variable actuation valve,        a second bushing slidably mounted within said first bushing and having one end in contact with a corresponding end of said actuating piston,        a first chamber defined between said bushing and said actuating piston, which is in communication with a passage for feeding the pressurised fluid to said first chamber,        a second chamber defined between said first bushing and said second bushing, and        a non return valve which controls a passage in a wall of said second bushing to allow the passage of fluid only from said first chamber to said second chamber of said auxiliary hydraulic tappet.        